Formalismo da integral funcional para dinâmica multiplicativa de Langevin

Abstract

We present a study on general multidimensional stochastic processes driven by a system of Langevin equations with multiplicative white noise. In particular, we address the problem of how time reversal difusion processes are affected by the variety of conventions available to deal with stochastic integrals. In the present thesis, we have studied a functional formalism used to construct correlation functions without any kind of discretization of the Langevin equations at any intermediate step. The generating functional is characterized by a functional integration over two sets of commuting variables, as well as Grassmann variables. The stochastic calculus is codified in our formalism in the structure of the Grassmann algebra. We have study some examples such as higher order derivative Langevin equations and the functional representation of the micromagnetic stochastic Landau-Lifshitz-Gilbert equation. With the help of functional formalism we calculate potentials for stationary nonequilibrium states, achieved by a multiplicative stochastic dynamics. we perform a weak-noise expansion, which allows the explicit evaluation of the potential in arbitrary dimensions and for any stochastic prescription. We apply this formalism to the study of noise-induced phase transitions, defining a local order parameter. By computing entropy production, we show that microscopic irreversibility is a necessary condition to develop noise-induced phase transitions. We apply our formalism to the study of conditional probability for a multiplicative Langevin dynamics. We propose the method of reparametrization that allows us to associate the problem of the variable mass of the quantum mechanics to the case of multiplicative noise. Finally we make an example that allows to validate the proposed method.